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 A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/27)^n follows from the series solutions of 2*T-d/dx(9*(1-x)*x*dT/dx)=0. 3
 0, 1, 15, 279, 5729, 124554, 2810718, 65114402, 1538182398, 36887880105, 895303119303, 21943398532563, 542209373589501, 13489931811324550, 337599511395854298, 8491805574767197650, 214548940430198454054, 5441921826542937659088, 138512110164878076019560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also appears in Ramanujan's theory of elliptic functions, signature 3 (cf. A006480). Almkvist et al. give a real and complex Ansatz for the second-order, ordinary differential equation: T_R = 1 + x*{Z[[x]]}, T_C = T_R*log(x) + x*{Z[[x]]}. REFERENCES B. C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82. LINKS G. Almkvist et al., Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275. [The article is on pages 273-295.] MATHEMATICA G[nMax_]:=Dot[RecurrenceTable[{Dot[{(3*n-5)^2 (3*n-4)^2 (9*n-4), -18(n - 1)(40 - 197*n + 351*n^2 - 279*n^3 + 81*n^4), 81(n - 1)*n^3*(9*n - 13)}, a[n-#] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 5/9}, a, {n, 0, nMax}], x^Range[0, nMax]]; qSer[nMax_] := Expand[Times[x, Normal[Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/3, 2/3, 1, x]]], {x, 0, nMax}]]]]; CoefficientList[(1/k)*qSer[20]/.{x->k*x}, x]/.{k->27} CROSSREFS Cf. A005797, A308836, A308837. Sequence in context: A034687 A159239 A279976 * A199096 A060542 A095654 Adjacent sequences:  A308832 A308833 A308834 * A308836 A308837 A308838 KEYWORD nonn AUTHOR Bradley Klee, Jun 27 2019 STATUS approved

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Last modified June 25 15:36 EDT 2022. Contains 354851 sequences. (Running on oeis4.)